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An sl(n) stable homotopy type for matched diagrams.

Jones, Dan and Lobb, Andrew and Schuetz, Dirk (2019) 'An sl(n) stable homotopy type for matched diagrams.', Advances in mathematics., 356 . p. 106816.


There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The Khovanov cohomology of a knot diagram made by gluing tangles of this type is therefore often amenable to calculation. We lift this idea to the level of the Lipshitz-Sarkar stable homotopy type and use it to make new computations. Similarly, there exists a simplified Khovanov-Rozansky sln complex for open 2-braids with oppositely oriented strands and an even number of crossings. Diagrams made by gluing tangles of this type are called matched diagrams, and knots admitting matched diagrams are called bipartite knots. To a pair consisting of a matched diagram and a choice of integer n ≥ 2, we associate a stable homotopy type. In the case n = 2 this agrees with the Lipshitz-Sarkar stable homotopy type of the underlying knot. In the case n ≥ 3 the cohomology of the stable homotopy type agrees with the sln Khovanov-Rozansky cohomology of the underlying knot. We make some consistency checks of this sln stable homotopy type and show that it exhibits interesting behaviour. For example we find a CP2 in the sl3 type for some diagram, and show that the sl4 type can be interesting for a diagram for which the Lipshitz-Sarkar type is a wedge of Moore spaces.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Publisher statement:© 2019 This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Date accepted:08 September 2019
Date deposited:12 September 2019
Date of first online publication:03 October 2019
Date first made open access:03 October 2020

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